Frobenius distributions in short intervals for CM elliptic curves

For an elliptic curve E/Q, Hasse's theorem asserts that #E(Fp)=p+1−ap, where |ap|≤2p. Assuming that E has complex multiplication, we establish asymptotics for primes p for which ap is in subintervals of the Hasse interval [−2p,2p] of measure o(p). In particular, given a function f=o(1) satisfying some mild conditions, we provide counting functions for primes p where |ap|∈(2p(1−f(p)),2p), and for primes where ap∈(2p(c−f(p)),2cp), where c∈(0,1) is a constant.